I am an undergrad and I know that the conjecture may have been proven recently. But in reading about it, I am entirely confused as to what it means and why it is important. I was hoping some of you kind people could help me.

I know there are several formulations of the conjecture.

Wolfram says:

for any infinitesimal $\epsilon > 0$, there exists a constant $C_\epsilon$ such that for any three relatively prime integers $a$, $b$, $c$ satisfying $a+b=c$ the inequality $$\max (|a|, |b|, |c|) \leq C_{\epsilon}\displaystyle\prod_{p|abc} p^{1+\epsilon}$$
holds, where $p|abc$ indicates that the product is over primes $p$ which divide the product $abc$.

Then Wikipedia says:

For a positive integer $n$, the radical of $n$, denoted $\text{rad}(n)$, is the product of the distinct prime factors of $n$. If $a$, $b$, and $c$ are coprime positive integers such that $a + b = c$, it turns out that "usually" $c < \text{rad}(abc)$. The abc conjecture deals with the exceptions. Specifically, it states that for every $\epsilon>0$ there exist only finitely many triples $(a,b,c)$ of positive coprime integers with $a + b = c$ such that $$c>\text{rad}(abc)^{1+\epsilon}$$

An equivalent formulation states that for any $\epsilon > 0$, there exists a constant $K$ such that, for all triples of coprime positive integers $(a, b, c)$ satisfying $a + b = c$, the inequality $$c<K\cdot\text{rad}(abc)^{1+\epsilon}$$

holds.

A third formulation of the conjecture involves the quality $q(a, b, c)$ of the triple $(a, b, c)$, defined by: $$q(a,b,c)=\frac{\log(c)}{\log(\text{rad}(abc)}$$

I am particularly interested in the first definition, but any help with any of it would be greatly appreciated.

@AndréNicolas I corrected the definition. Also, I agree about the infinitesimals, I just assume it is an $\epsilon$ like in an analysis proof. It is just a positive number that we usually consider to be small, but it can truly any positive number. Is that right?
–
Joseph SkeltonSep 15 '12 at 4:46

Yes, what they should have said is that for any $\epsilon \gt 0$, there exists a $C_{\epsilon}$ such that $\dots$. The Wikipedia article is pretty good.
–
André NicolasSep 15 '12 at 5:20

$C_{\epsilon }$ is missing on the RHS of the first inequality (from WolframMathWorld).
–
Américo TavaresSep 16 '12 at 21:54

so far, five answers, but nobody tried to explain why the abc conjecture is considered important!
–
kjetil b halvorsenOct 21 '12 at 3:25

In Serge Lang's Algebra, he says: "One of the most fruitful analogies in mathematics is that between the integers and the ring of polynomials over a field". He then proves the abc conjecture for polynomials, and for good measure he proves Fermat's Last Theorem for polynomials. In other words, Lang is saying that if something is true for the ring of polynomials, one ought to check if it is true for that rather important ring called the integers. But it turns out that the ring of integers can be rather more troublesome, which may be surprising. So I'd say the abc conjecture is important because its proof over polynomial rings tells you it ought to be true for integers, but like Fermat it is rather more elusive than it appears. if you have access to Lang, his writeup in Chapter IV.7 is really good.

Suppose a, b, and c are coprime positive integers: $0<a<b<c=a+b$ with $\gcd(a,b)=\gcd(a,c)=\gcd(b,c)=1.$ Then (under the abc conjecture) there are only finitely many such a, b, and c such that $c>\operatorname{rad}(abc)^{1.1}$, only finitely many such that $c>\operatorname{rad}(abc)^{1.01}$, only finitely many such that $c>\operatorname{rad}(abc)^{1.001}$, etc.

Another way: let $p_1,p_2,\ldots,p_k$ be the set of primes dividing $abc$ with exponents $a_1,\ldots,a_k,b_1,\ldots,c_k$ ($\min(a_i,b_i,c_i)\ge0$ and $\max(a_i,b_i,c_i)\ge1$ for all $i$). Then
$$
c=p_1^{c_1}p_2^{c_2}\cdots p_k^{c_k}>(p_1p_2\cdots p_k)^{1.001}
$$
only finitely often (where 1.001 can be replaced with any number greater than 1).